Theorem bezout 15260

Description: Bézout's identity: For any integers A and B, there are integers x , y such that ( A gcd B ) = A x. x + B x. y. This is Metamath 100 proof #60. (Contributed by Mario Carneiro, 22-Feb-2014.)

Assertion

Ref	Expression
bezout	|- ( ( A e. ZZ /\ B e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem bezout

Dummy variables t u v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . . . . 8 |- ( z = t -> ( z = ( ( A x. x ) + ( B x. y ) ) <-> t = ( ( A x. x ) + ( B x. y ) ) ) )
2	1	2rexbidv 3057	. . . . . . 7 |- ( z = t -> ( E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) <-> E. x e. ZZ E. y e. ZZ t = ( ( A x. x ) + ( B x. y ) ) ) )
3		oveq2 6658	. . . . . . . . . 10 |- ( x = u -> ( A x. x ) = ( A x. u ) )
4	3	oveq1d 6665	. . . . . . . . 9 |- ( x = u -> ( ( A x. x ) + ( B x. y ) ) = ( ( A x. u ) + ( B x. y ) ) )
5	4	eqeq2d 2632	. . . . . . . 8 |- ( x = u -> ( t = ( ( A x. x ) + ( B x. y ) ) <-> t = ( ( A x. u ) + ( B x. y ) ) ) )
6		oveq2 6658	. . . . . . . . . 10 |- ( y = v -> ( B x. y ) = ( B x. v ) )
7	6	oveq2d 6666	. . . . . . . . 9 |- ( y = v -> ( ( A x. u ) + ( B x. y ) ) = ( ( A x. u ) + ( B x. v ) ) )
8	7	eqeq2d 2632	. . . . . . . 8 |- ( y = v -> ( t = ( ( A x. u ) + ( B x. y ) ) <-> t = ( ( A x. u ) + ( B x. v ) ) ) )
9	5, 8	cbvrex2v 3180	. . . . . . 7 |- ( E. x e. ZZ E. y e. ZZ t = ( ( A x. x ) + ( B x. y ) ) <-> E. u e. ZZ E. v e. ZZ t = ( ( A x. u ) + ( B x. v ) ) )
10	2, 9	syl6bb 276	. . . . . 6 |- ( z = t -> ( E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) <-> E. u e. ZZ E. v e. ZZ t = ( ( A x. u ) + ( B x. v ) ) ) )
11	10	cbvrabv 3199	. . . . 5 |- { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) } = { t e. NN | E. u e. ZZ E. v e. ZZ t = ( ( A x. u ) + ( B x. v ) ) }
12		simpll 790	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. ZZ )
13		simplr 792	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> B e. ZZ )
14		eqid 2622	. . . . 5 |- inf ( { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) } , RR , < ) = inf ( { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) } , RR , < )
15		simpr 477	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> -. ( A = 0 /\ B = 0 ) )
16	11, 12, 13, 14, 15	bezoutlem4 15259	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) } )
17		eqeq1 2626	. . . . . . 7 |- ( z = ( A gcd B ) -> ( z = ( ( A x. x ) + ( B x. y ) ) <-> ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) ) )
18	17	2rexbidv 3057	. . . . . 6 |- ( z = ( A gcd B ) -> ( E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) <-> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) ) )
19	18	elrab 3363	. . . . 5 |- ( ( A gcd B ) e. { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) } <-> ( ( A gcd B ) e. NN /\ E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) ) )
20	19	simprbi 480	. . . 4 |- ( ( A gcd B ) e. { z e. NN | E. x e. ZZ E. y e. ZZ z = ( ( A x. x ) + ( B x. y ) ) } -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )
21	16, 20	syl 17	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )
22	21	ex 450	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( -. ( A = 0 /\ B = 0 ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) ) )
23		0z 11388	. . . 4 |- 0 e. ZZ
24		00id 10211	. . . . 5 |- ( 0 + 0 ) = 0
25		0cn 10032	. . . . . . 7 |- 0 e. CC
26	25	mul01i 10226	. . . . . 6 |- ( 0 x. 0 ) = 0
27	26, 26	oveq12i 6662	. . . . 5 |- ( ( 0 x. 0 ) + ( 0 x. 0 ) ) = ( 0 + 0 )
28		gcd0val 15219	. . . . 5 |- ( 0 gcd 0 ) = 0
29	24, 27, 28	3eqtr4ri 2655	. . . 4 |- ( 0 gcd 0 ) = ( ( 0 x. 0 ) + ( 0 x. 0 ) )
30		oveq2 6658	. . . . . . 7 |- ( x = 0 -> ( 0 x. x ) = ( 0 x. 0 ) )
31	30	oveq1d 6665	. . . . . 6 |- ( x = 0 -> ( ( 0 x. x ) + ( 0 x. y ) ) = ( ( 0 x. 0 ) + ( 0 x. y ) ) )
32	31	eqeq2d 2632	. . . . 5 |- ( x = 0 -> ( ( 0 gcd 0 ) = ( ( 0 x. x ) + ( 0 x. y ) ) <-> ( 0 gcd 0 ) = ( ( 0 x. 0 ) + ( 0 x. y ) ) ) )
33		oveq2 6658	. . . . . . 7 |- ( y = 0 -> ( 0 x. y ) = ( 0 x. 0 ) )
34	33	oveq2d 6666	. . . . . 6 |- ( y = 0 -> ( ( 0 x. 0 ) + ( 0 x. y ) ) = ( ( 0 x. 0 ) + ( 0 x. 0 ) ) )
35	34	eqeq2d 2632	. . . . 5 |- ( y = 0 -> ( ( 0 gcd 0 ) = ( ( 0 x. 0 ) + ( 0 x. y ) ) <-> ( 0 gcd 0 ) = ( ( 0 x. 0 ) + ( 0 x. 0 ) ) ) )
36	32, 35	rspc2ev 3324	. . . 4 |- ( ( 0 e. ZZ /\ 0 e. ZZ /\ ( 0 gcd 0 ) = ( ( 0 x. 0 ) + ( 0 x. 0 ) ) ) -> E. x e. ZZ E. y e. ZZ ( 0 gcd 0 ) = ( ( 0 x. x ) + ( 0 x. y ) ) )
37	23, 23, 29, 36	mp3an 1424	. . 3 |- E. x e. ZZ E. y e. ZZ ( 0 gcd 0 ) = ( ( 0 x. x ) + ( 0 x. y ) )
38		oveq12 6659	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
39		oveq1 6657	. . . . . 6 |- ( A = 0 -> ( A x. x ) = ( 0 x. x ) )
40		oveq1 6657	. . . . . 6 |- ( B = 0 -> ( B x. y ) = ( 0 x. y ) )
41	39, 40	oveqan12d 6669	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( ( A x. x ) + ( B x. y ) ) = ( ( 0 x. x ) + ( 0 x. y ) ) )
42	38, 41	eqeq12d 2637	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) <-> ( 0 gcd 0 ) = ( ( 0 x. x ) + ( 0 x. y ) ) ) )
43	42	2rexbidv 3057	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) <-> E. x e. ZZ E. y e. ZZ ( 0 gcd 0 ) = ( ( 0 x. x ) + ( 0 x. y ) ) ) )
44	37, 43	mpbiri 248	. 2 |- ( ( A = 0 /\ B = 0 ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )
45	22, 44	pm2.61d2 172	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   NNcn 11020   ZZcz 11377   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  dvdsgcd  15261  dvdsmulgcd  15274  lcmgcdlem  15319  divgcdcoprm0  15379  odbezout  17975  ablfacrp  18465  pgpfac1lem3  18476  znunit  19912  2sqb  25157  ostth3  25327